Officers are looking for any link between the “explosive device” and the message – also sent to the Evening News.
No one was hurt when army bomb disposal experts carried out a controlled explosion on the shoebox after it was left in a public shelter.
A Police Scotland spokesman said: “We have received a number of e-mail correspondence since our public appeal regarding the suspicious package found in Princes Street Gardens on Thursday, January 11, 2018.
“Detectives are actively pursuing various lines of inquiry at this time.
“Our message to communities remains the same, however; be alert, not alarmed and if you witness any suspicious activity across the country, contact police immediately.
“We also continue to ask members of the public with information relating to the suspicious package in Princes Street Gardens to get in touch.”
The Evening News received the message via a US-based e-mail service designed to protect the anonymity of users by withholding internet addresses.
A link within the e-mail is to an extremist website which references the January 11 discovery and includes a photograph purporting to be the device prior to detonation.
The group named in the e-mail and on the website has no links to Islamic terrorism.
Initially thought to be an “elaborate hoax”, police confirmed last month that the contraption had the “potential to cause injury”. A park ranger found the box and immediately called police – triggering emergency protocols.
Army bomb disposal experts from Craigiehall were scrambled and carried out a controlled detonation on the package about three hours after the alarm was first raised.
A “Gold” meeting of senior commanders decided to extend the parameters of the investigation and call in specialist support to conduct a sweep of the area.
Counter-terrorism police were also brought in while Scottish ministers were kept informed. A trawl of CCTV to identify who left the package proved unsuccessful.
Officers said at the time that no note was left with the package and no phone call warning was made to emergency services.
Passers-by were in the grounds at the time, but no evacuation was carried out.
The package – understood to contain wires and a battery – is still undergoing tests but police refused to give further details about its contents and capability.
Detective Chief Inspector Kenny Armstrong said: “I’d like to hear from anyone who saw any suspicious behaviour in the immediate vicinity of the gardens or who has information as to who may be responsible.”
Anyone with information can contact officers via 101 quoting incident number 1204 of February 21 or anonymously through Crimestoppers on 0800 555 111.